Question

Consider the following expression.
-2x(m + n - 4) + 5(-2m + 2n) + n(m + 4n - 5)
Which of the following is an equivalent expression?
-10m^2 + 4n^2 - mn - 2m + 5n
-10m^2 + 4n^2 - 3mn - 2m + 5n
-2^2 + 4n^2 - mn - 2m + 5n
-2n^2 + 4n^2 - 3mn 2m + 5n

Answer 1

yes I would explore this further

Answer 2

Let's simplify the given expression step by step:

The expression is:
\[
-2x(m + n - 4) + 5(-2m + 2n) + n(m + 4n - 5)
\]

Since there's no \( x \) present when evaluating the other terms, we can ignore the \(-2x(m + n - 4)\) part for simplification purposes in this context. Let's simplify the other parts first.

### Step 1: Simplify each term

1. **First term**:
\[
5(-2m + 2n) = -10m + 10n
\]

2. **Second term**:
\[
n(m + 4n - 5) = nm + 4n^2 - 5n
\]

### Step 2: Combine all terms

Now we combine:
- Ignore the \(-2x(m+n-4)\): It's not affecting the other components.
- From the first term, we have \(-10m + 10n\).
- From the second term, we have \(nm + 4n^2 - 5n\).

So, we combine:
\[
-10m + 10n + nm + 4n^2 - 5n
\]

### Step 3: Combine like terms

Now combine the coefficients:
- The \(n\) terms: \(10n - 5n = 5n\)
- The expression simplifies to:
\[
-10m + nm + 4n^2 + 5n
\]

### Step 4: Rearranging

Write it in a conventional form:
\[
-10m + nm + 5n + 4n^2
\]
which can also be written as:
\[
-10m + nm + 4n^2 + 5n
\]

### Step 5: Identifying equivalent expressions

Now let's examine the given options:

1. \(-10m^2 + 4n^2 - mn - 2m + 5n\)
2. \(-10m^2 + 4n^2 - 3mn - 2m + 5n\)
3. \(-2^2 + 4n^2 - mn - 2m + 5n\)
4. \(-2n^2 + 4n^2 - 3mn + 2m + 5n\)

None of the options match our derived expression of:
\[
nm + 4n^2 - 10m + 5n
\]
Let's note that we have \(mn\) rather than \(-mn\), \( -10m\) rather than \( -10m^2\), and so forth.

After re-verifying the individual terms and simplifications, none of these options properly correspond to the derived expression. Double-checking or additional context might be necessary to resolve this. Would you like to explore this further?

Answer 3

Let's take a more methodical approach to simplifying the original expression:

### Given Expression:
\[
-2x(m + n - 4) + 5(-2m + 2n) + n(m + 4n - 5)
\]

### Step 1: Expand Each Term

1. The first term \(-2x(m + n - 4)\) can be left as is for now since \(x\) is not a variable we're focusing on for simplification.

2. The second term:
\[
5(-2m + 2n) = 5 \cdot -2m + 5 \cdot 2n = -10m + 10n
\]

3. The third term:
\[
n(m + 4n - 5) = nm + 4n^2 - 5n
\]

### Step 2: Combine All Terms (Ignoring the First)

Now, let's combine the terms from the second and third expansions:
\[
-10m + 10n + nm + 4n^2 - 5n
\]

### Step 3: Combine Like Terms

Now, let's combine like terms:
- The \(n\) terms: \(10n - 5n = 5n\)

Putting it all together, we have:
\[
nm - 10m + 4n^2 + 5n
\]

This simplifies to:
\[
mn - 10m + 4n^2 + 5n
\]
(Here, \(mn\) is the same as \(nm\); the order of multiplication doesn't matter.)

### Step 4: Identifying Equivalent Expressions

Now we have:
\[
mn - 10m + 4n^2 + 5n
\]

Next, let's examine the given options once again:

1. \(-10m^2 + 4n^2 - mn - 2m + 5n\)
2. \(-10m^2 + 4n^2 - 3mn - 2m + 5n\)
3. \(-2^2 + 4n^2 - mn - 2m + 5n\)
4. \(-2n^2 + 4n^2 - 3mn + 2m + 5n\)

### Analyzing the Options

None of the options are identical to \(mn - 10m + 4n^2 + 5n\). Let’s analyze this more closely:

**1. \(-10m^2 + 4n^2 - mn - 2m + 5n\)**:
- This has \(-10m^2\) instead of \(-10m\), so it doesn't match because we have linear \(m\) terms, not quadratic.

**2. \(-10m^2 + 4n^2 - 3mn - 2m + 5n\)**:
- Again, it has \(-10m^2\) and a different \(mn\) term. Not a match.

**3. \(-2^2 + 4n^2 - mn - 2m + 5n\)**:
- This doesn't match due to \(-2^2\), which evaluates to \(-4\).

**4. \(-2n^2 + 4n^2 - 3mn + 2m + 5n\)**:
- This has different signs and different coefficients.

### Conclusion

None of the provided options seem to fit our derived expression of:
\[
mn - 10m + 4n^2 + 5n
\]
If we re-examine the problem or if I misunderstood the involvement of \(x\), please double-check the context or the exact formulation of the question presented.

If you want to further explore how we could potentially match one of these options, please let me know how you would like to proceed!